Final answer:
The nervous system with postganglionic fibers that do not innervate cutaneous effectors in skin or skeletal muscles is the sympathetic nervous system. The sympathetic system can release ACh through its postganglionic fibers in certain cases, and also influences systemic effects via the adrenal medulla.
Step-by-step explanation:
The student's question asks which part of the nervous system has postganglionic fibers that do not travel in spinal nerves and therefore do not innervate cutaneous effectors in skin or skeletal muscles. The correct answer here is the sympathetic nervous system. Postganglionic fibers in the sympathetic nervous system can release acetylcholine (ACh) instead of norepinephrine in certain cases such as contacting blood vessels within skeletal muscle or contacting sweat glands. Notably, the parasympathetic nervous system typically innervates organs via very short postganglionic fibers that travel alongside or within the organ itself.
The sympathetic system also includes specialized preganglionic connections to the adrenal medulla that release epinephrine and norepinephrine into the bloodstream, which can cause widespread systemic effects. It's important to recognize the sympathetic chain ganglia are adjacent to the vertebral column and thus the preganglionic sympathetic fibers are relatively short and myelinated, while postganglionic sympathetic fibers are longer and unmyelinated due to the greater distance to the target effector.